Unbalance in rotating machines is
a common source of vibration excitation. We consider here a spring-mass system constrained
to move in the vertical direction and excited by a rotating machine that is unbalanced, as
shown in Fig. 10. The unbalance is represented by an eccentric mass m with
eccentricity e that is rotating with angular velocity w . By letting x be
the displacement of the non rotating mass (M - m) from the static equilibrium
position, the displacement of m is :

Figure 10 Harmonic
Disturbing Force Resulting from Rotating Unbalance

The equation of motion is then :

which can be rearranged to :

(37)

It is evident that this equation is
identical to Eq. (29), where is
replaced by , and hence the steady-state solution of the
previous section can be replaced by :

(38)

and

(39)

These can be further reduced to non
dimensional form :

(40)

and

(41)

Example

A counter rotating eccentric
weight exciter is used to produce the forced oscillation of a spring-supported mass as
shown in Fig. 11. By varying the speed of rotation, a resonant amplitude of 0.60 cm was
recorded. When the speed of rotation was increase considerably beyond the resonant
frequency, the amplitude appeared to approach a fixed value of 0.08 cm. Determine the
damping factor of the system.

Figure 11

Solution :

From Eqn. (40), the resonant amplitude
is :

When w is very much greater than , the same equation becomes

By solving the two equations
simultaneously, the damping factor of the system is